Question

How do you use a Riemann Sum with n = 4 to estimate `ln3 = int (1/x)` from 1 to 3 using the right endpoints and then the midpoints?

Answer 1

Riemann Sum with n = 4 to estimate `int (1/x)` from 1 to 3 using the right endpoints and then the midpoints

`f(x) = 1/x`

`[a,b] = [1,3]`

`n=4`

`Delta x = (b-a)/n = (3-1)/4 = 1/2`

The endpoints of the subintervals are found by beginning at `a` and successively adding `Deltax` until we get to `b`.

The subintervals are:

`[1,3/2] [3/2,2] [2,5/2] [5/2,3]`

The right endpoints are `x_1=3/2, x_2-2, x_3=5/2, x_4=3`

`R_4 = f(x_1)Deltax + f(x_2)Deltax+ f(x_3)Deltax+ f(x_4)Deltax`

Plug in the numbers and do the arithmetic.

The midpoints of the intervals can be found by averaging the endpoints or by finding the first midpoint and successively adding `Deltax`

`m_1 = 5/4, m_2 = 7/4, m_3 = 9/4, m_4=11/4`

`M_4 = f(m_1)Deltax + f(m_2)Deltax+ f(m_3)Deltax+ f(m_4)Deltax`

Plug in the numbers and do the arithmetic